3704 summary

Although it’s still quite incomplete, I’ve made available on the course website a preliminary summary of the material you should review for the exams. I’m likely to modify and expand it, especially if I receive comments, so you should check back every now and then. Happy Christmas and New Year! MK

2201 Summary

Although it’s still quite incomplete, I’ve made available on the course website a preliminary summary of the material you should review for the exams. I’m likely to modify and expand it, especially if I receive comments, so you should check back every now and then. Happy Christmas and New Year! MK

Non-commutative constructions

In June of next year, there will be a small workshop at UCL with the title `non-commutative constructions in arithmetic and geometry.’ I will write later in more detail about the goals of the meeting, but here is a link to a preliminary web page. MK

Mathematical form

During a lecture on linear algebra, I was giving a loose description of orthonormal bases in the space of vibrational modes of a metal string. It seemed noteworthy that the underlying vector space was infinite-dimensional, but was approximated by finite-dimensional spaces in practice, corresponding to ignoring high frequencies. That is to say, it is a [...]

Fundamental groups

Yesterday, at the last algebraic number theory lecture of the term, I defined the fundamental group \pi_1(O_K) of the ring of algebraic integers O_K in an algebraic number field K as the Galois group Gal(F/K), where F is the compositum of all the unramified extensions of K. And then, it was stated that the ideal [...]

Isometries in dimension 4?

It occurs to me that I’ve never thought about higher-dimensional generalizations of the two theorems contained in the previous post, say even in dimension 4. I’m sure it’s well-known in some form, but by thinking about it, you might even find a nice new description! To be precise, we are asking for a natural description [...]

Isometries in two and three dimensions

The 2201 course website for linear algebra now has a write-up of the last lecture. The note on self-adjoint maps has been corrected and somewhat revised. MK

Spec(Z) and three-manifolds

The following question comes from an old friend, John Baez. John is a renowned mathematical physicist and internet guru. You should look at his website and his blog for an enormously informative and entertaining survey of mathematics and physics. My reply is here. MK Hi! Why is Spec(Z) 3-dimensional? I think I understand what the [...]

Self-adoint linear maps

The course website has a very superficial note on self-adjoint maps to complement the lecture of 13/12/2007. MK

Mathematical vistas

My first term in the UK and some recent contact with issues in primary and secondary education prompted a brief reflection on the teaching of mathematics to children, written mostly with fellow parents in mind. But if you have the time to read it, I would very much welcome comments from maths students as well. [...]